Approximations of the tail index estimator of heavy-tailed distributions under random censoring and application

We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation, with the same variance as that obtained by Einmahl, Fils-Villetard and Guillou (2008, Bernoulli 14, no. 1, 207-227). The newly proposed Gaussian approximation agrees perfectly with the asymptotic representation of the classical Hill estimator in the non censoring framework. Our results will be of great interest to establish the limit distributions of many statistics in extreme value theory under random censoring such as the estimators of tail indices, the actuarial risk measures and the goodness-of-fit functionals for heavy-tailed distributions. As an application, we establish the asymptotic normality of an estimator of the excess-of-loss reinsurance premium.